Future value Present value Rates of return Amortization Time Value of Money

Time lines show timing of cash flows. CF 0 CF 1 CF 3 CF i% Tick marks at ends of periods, so Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2.

Time line for a $100 lump sum due at the end of Year Year i%

Time line for an ordinary annuity of $100 for 3 years i%

Time line for uneven CFs -$50 at t = 0 and $100, $75, and $50 at the end of Years 1 through i% -50

What’s the FV of an initial $100 after 3 years if i = 10%? FV = ? % 100 Finding FVs is compounding.

After 1 year: FV 1 = PV + INT 1 = PV + PV(i) = PV(1 + i) = $100(1.10) = $ After 2 years: FV 2 = PV(1 + i) 2 = $100(1.10) 2 = $

After 3 years: FV 3 = PV(1 + i) 3 = 100(1.10) 3 = $ In general, FV n = PV(1 + i) n.

Four Ways to Find FVs Solve the equation with a regular calculator. Use tables. Use a financial calculator. Use a spreadsheet.

Financial calculators solve this equation: FV n = PV(1 + i) n. There are 4 variables. If 3 are known, the calculator will solve for the 4th. Financial Calculator Solution

Here’s the setup to find FV: Clearing automatically sets everything to 0, but for safety enter PMT = 0. Set:P/YR = 1, END INPUTS OUTPUT NI/YR PV PMT FV

10% What’s the PV of $100 due in 3 years if i = 10%? Finding PVs is discounting, and it’s the reverse of compounding PV = ?

Solve FV n = PV(1 + i ) n for PV:   PV= $ = $100PVIF = $ = $ i,n       3.

Financial Calculator Solution N I/YR PV PMTFV Either PV or FV must be negative. Here PV = Put in $75.13 today, take out $100 after 3 years. INPUTS OUTPUT

If sales grow at 20% per year, how long before sales double? Solve for n: FV n = 1(1 + i) n ; 2 = 1(1.20) n Use calculator to solve, see next slide.

N I/YR PV PMTFV 3.8 Graphical Illustration: FV 3.8 Year INPUTS OUTPUT

Ordinary Annuity PMT 0123 i% PMT 0123 i% PMT Annuity Due What’s the difference between an ordinary annuity and an annuity due?

What’s the FV of a 3-year ordinary annuity of $100 at 10%? % FV= 331

Financial Calculator Solution Have payments but no lump sum PV, so enter 0 for present value. INPUTS OUTPUT I/YRNPMTFVPV

What’s the PV of this ordinary annuity? % = PV

Have payments but no lump sum FV, so enter 0 for future value INPUTS OUTPUT NI/YRPVPMTFV

Find the FV and PV if the annuity were an annuity due % 100

Switch from “End” to “Begin.” Then enter variables to find PVA 3 = $ Then enter PV = 0 and press FV to find FV = $ INPUTS OUTPUT NI/YRPVPMTFV

What is the PV of this uneven cash flow stream? % = PV

Input in “CFLO” register: CF 0 = 0 CF 1 = 100 CF 2 = 300 CF 3 = 300 CF 4 = -50 Enter I = 10, then press NPV button to get NPV = (Here NPV = PV.)

What interest rate would cause $100 to grow to $ in 3 years? % $100 (1 + i ) 3 = $ INPUTS OUTPUT NI/YRPVPMTFV

Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated I% constant? Why? LARGER! If compounding is more frequent than once a year--for example, semiannually, quarterly, or daily--interest is earned on interest more often.

% % Annually: FV 3 = 100(1.10) 3 = Semiannually: FV 6 = 100(1.05) 6 =

We will deal with 3 different rates: i Nom = nominal, or stated, or quoted, rate per year. i Per = periodic rate. EAR= EFF% =. effective annual rate

i Nom is stated in contracts. Periods per year (m) must also be given. Examples: l 8%; Quarterly l 8%, Daily interest (365 days)

Periodic rate = i Per = i Nom /m, where m is number of compounding periods per year. m = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding. Examples: 8% quarterly: i Per = 8%/4 = 2%. 8% daily (365): i Per = 8%/365 = %.

Effective Annual Rate (EAR = EFF%): The annual rate that causes PV to grow to the same FV as under multi-period compounding. Example: EFF% for 10%, semiannual: FV = (1 + i Nom /m) m = (1.05) 2 = EFF% = 10.25% because (1.1025) 1 = Any PV would grow to same FV at 10.25% annually or 10% semiannually.

An investment with monthly payments is different from one with quarterly payments. Must put on EFF% basis to compare rates of return. Use EFF% only for comparisons. Banks say “interest paid daily.” Same as compounded daily.

How do we find EFF% for a nominal rate of 10%, compounded semiannually? Or use a financial calculator.

EAR = EFF% of 10% EAR Annual = 10%. EAR Q =( /4) 4 – 1= 10.38%. EAR M =( /12) 12 – 1= 10.47%. EAR D(360) =( /360) 360 – 1= 10.52%.

Can the effective rate ever be equal to the nominal rate? Yes, but only if annual compounding is used, i.e., if m = 1. If m > 1, EFF% will always be greater than the nominal rate.

When is each rate used? i Nom :Written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines.

i Per : Used in calculations, shown on time lines. If i Nom has annual compounding, then i Per = i Nom /1 = i Nom.

(Used for calculations if and only if dealing with annuities where payments don’t match interest compounding periods.) EAR = EFF%: Used to compare returns on investments with different payments per year.

FV of $100 after 3 years under 10% semiannual compounding? Quarterly? = $100(1.05) 6 = $ FV 3Q = $100(1.025) 12 = $ FV = PV1.+ i m n Nom mn       FV = $ S 2x3      

What’s the value at the end of Year 3 of the following CF stream if the quoted interest rate is 10%, compounded semiannually? % mos. periods 100

Payments occur annually, but compounding occurs each 6 months. So we can’t use normal annuity valuation techniques.

1st Method: Compound Each CF % FVA 3 = 100(1.05) (1.05) =

Could you find FV with a financial calculator? Yes, by following these steps: a. Find the EAR for the quoted rate: 2nd Method: Treat as an Annuity EAR = ( 1 + ) – 1 = 10.25%

Or, to find EAR with a calculator: NOM% = 10. P/YR = 2. EFF% =

EFF% = P/YR = 1 NOM% = INPUTS OUTPUT NI/YRPVFVPMT b. The cash flow stream is an annual annuity. Find k Nom (annual) whose EFF% = 10.25%. In calculator, c.

What’s the PV of this stream? %

Amortization Construct an amortization schedule for a $1,000, 10% annual rate loan with 3 equal payments.

Step 1: Find the required payments. PMT % -1, INPUTS OUTPUT NI/YRPVFVPMT

Step 2: Find interest charge for Year 1. INT t = Beg bal t (i) INT 1 = $1,000(0.10) = $100. Step 3: Find repayment of principal in Year 1. Repmt = PMT – INT = $ – $100 = $

Step 4: Find ending balance after Year 1. End bal = Beg bal – Repmt = $1,000 – $ = $ Repeat these steps for Years 2 and 3 to complete the amortization table.

Interest declines. Tax implications. BEGPRINEND YRBALPMTINTPMTBAL 1$1,000$402$100$302$ TOT1, ,000

$ Interest Level payments. Interest declines because outstanding balance declines. Lender earns 10% on loan outstanding, which is falling. Principal Payments

Amortization tables are widely used- -for home mortgages, auto loans, business loans, retirement plans, etc. They are very important! Financial calculators (and spreadsheets) are great for setting up amortization tables.

On January 1 you deposit $100 in an account that pays a nominal interest rate of 10%, with daily compounding (365 days). How much will you have on October 1, or after 9 months (273 days)? (Days given.)

i Per = 10.0% / 365 = % per day. FV = ? % -100 Note: % in calculator, decimal in equation.   FV = $ = $ = $

INPUTS OUTPUT NI/YRPVFVPMT i Per =i Nom /m =10.0/365 = % per day. Enter i in one step. Leave data in calculator.

Now suppose you leave your money in the bank for 21 months, which is 1.75 years or = 638 days. How much will be in your account at maturity? Answer:Override N = 273 with N = 638. FV = $

i Per = % per day. FV = days -100 FV=$100(1 +.10/365) 638 =$100( ) 638 =$100(1.1910) =$

You are offered a note that pays $1,000 in 15 months (or 456 days) for $850. You have $850 in a bank that pays a 7.0% nominal rate, with 365 daily compounding, which is a daily rate of % and an EAR of 7.25%. You plan to leave the money in the bank if you don’t buy the note. The note is riskless. Should you buy it?

3 Ways to Solve: 1. Greatest future wealth: FV 2. Greatest wealth today: PV 3. Highest rate of return: Highest EFF% i Per = % per day. 1, days

1. Greatest Future Wealth Find FV of $850 left in bank for 15 months and compare with note’s FV = $1,000. FV Bank = $850( ) 456 = $ in bank. Buy the note: $1,000 > $

INPUTS OUTPUT NI/YRPVFVPMT Calculator Solution to FV: i Per =i Nom /m =7.0/365 = % per day. Enter i Per in one step.

2. Greatest Present Wealth Find PV of note, and compare with its $850 cost: PV=$1,000/( ) 456 =$

INPUTS OUTPUT NI/YRPVFV 7/365 = PV of note is greater than its $850 cost, so buy the note. Raises your wealth. PMT

Find the EFF% on note and compare with 7.25% bank pays, which is your opportunity cost of capital: FV n = PV(1 + i) n $1,000 = $850(1 + i) 456 Now we must solve for i. 3. Rate of Return

% per day INPUTS OUTPUT NI/YRPVFVPMT Convert % to decimal: Decimal = /100 = EAR = EFF%= ( ) 365 – 1 = 13.89%.

Using interest conversion: P/YR = 365. NOM% = (365) = EFF% = Since 13.89% > 7.25% opportunity cost, buy the note.